Theorem sepnsepolem2 49178

Description: Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 49179. Proof could be shortened by 1 step using ssdisjdr 49064. (Contributed by Zhi Wang, 1-Sep-2024.)

Hypothesis

Ref	Expression
sepnsepolem2.1	⊢ (𝜑 → 𝐽 ∈ Top)

Assertion

Ref	Expression
sepnsepolem2	⊢ (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))

Distinct variable groups:   𝑦,𝐷   𝑦,𝐽   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐷(𝑥)   𝐽(𝑥)

Proof of Theorem sepnsepolem2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sepnsepolem2.1	. 2 ⊢ (𝜑 → 𝐽 ∈ Top)
2		id 22	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 ∈ Top)
3		sslin 4195	. . . . 5 ⊢ (𝑧 ⊆ 𝑦 → (𝑥 ∩ 𝑧) ⊆ (𝑥 ∩ 𝑦))
4		sseq0 4355	. . . . . 6 ⊢ (((𝑥 ∩ 𝑧) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∩ 𝑧) = ∅)
5	4	ex 412	. . . . 5 ⊢ ((𝑥 ∩ 𝑧) ⊆ (𝑥 ∩ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑧) = ∅))
6	3, 5	syl 17	. . . 4 ⊢ (𝑧 ⊆ 𝑦 → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑧) = ∅))
7	6	adantl 481	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑧) = ∅))
8		ineq2 4166	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝑧))
9	8	eqeq1d 2738	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑧) = ∅))
10	9	adantl 481	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑦 = 𝑧) → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑧) = ∅))
11	2, 7, 10	opnneieqv 49166	. 2 ⊢ (𝐽 ∈ Top → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
12	1, 11	syl 17	1 ⊢ (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  ‘cfv 6492  Topctop 22837  neicnei 23041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-top 22838  df-nei 23042

This theorem is referenced by:  sepnsepo  49179